> that would make your storage medium a potential biohazard
generally no. If you're worried about random DNA being a biohazard there are way worse things to worry about, like how your immune system uses random stretches of biologically primed dna to create antibody diversity.
The real reasons why it's terrible is that write speed is atrocious and read speed is bad (on the order of 2-3x that of amazon glacier's robotic tape handlers, with WAY MORE expensive robots, and way more expensive cost to read -- you're bulk polluting rivers in china to make the reagents).
The only use case I can think of is deep generational archival (like the svalbard seed bank, but for information). Where cost to store by volume is at a premium, and where you'd like to have many many many copies, and you don't mind the cost to read because you won't be reading it but for every 10 or so years, if even.
Store your logs in DNA. You're never going to read them anyways.